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Extreme Events, Heavy Tails, and the Generating Processes: Examples from Hydrology and Geomorphology

Extreme Events, Heavy Tails, and the Generating Processes: Examples from Hydrology and Geomorphology. Efi Foufoula-Georgiou SAFL, NCED University of Minnesota. E2C2 – GIACS Advanced School on “Extreme Events: Nonlinear Dynamics and Time Series Analysis Comorova, Romania

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Extreme Events, Heavy Tails, and the Generating Processes: Examples from Hydrology and Geomorphology

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  1. Extreme Events, Heavy Tails, and the Generating Processes: Examples from Hydrology and Geomorphology Efi Foufoula-Georgiou SAFL, NCED University of Minnesota E2C2 – GIACS Advanced School on “Extreme Events: Nonlinear Dynamics and Time Series Analysis Comorova, Romania September 3-11, 2007

  2. Underlying Theme • In Hydrology and Geomorphology “Fluctuations” around the mean behavior are of high magnitude. • Understanding their statistical behavior is useful for prediction of extremes and also for understanding spatio-temporal heterogeneities which are hallmarks of the underlying process- generating mechanism. • These fluctuations are often found to exhibit power law tails and scaling

  3. PRESENCE OF SCALING ... scaling laws never appear by accident. They always manifest a property of the phenomenon of basic importance … This behavior should be discovered, if it exists, and its absence should also be recognized.” Barenblatt (2003)

  4. ~ 5 hrs ~ 1 hr t = 10s t = 5s High-resolution temporal rainfall data (courtesy, Iowa Institute of Hydraulic Research – IIHR)

  5. Noyo River basin

  6. STREAMLAB 2006 • Data Available: • Sediment accumulation series • Time series of bed elevation • Laser transects of bed elevation Pan-2 Pan-3 Pan-4 Pan-1 Pan-5

  7. Bed Elevation

  8. Noise-free sediment transport rates Weigh pan bedload transport rates (Q = 5.5 m3/s) (a) 1 s averaging and 0 point skip (b) 15s averaging time and 6 point skip (from Ramooz and Rennie, 2007)

  9. Localized Scaling Analysis: Multifractal Formalism • Characterize a signal f(x) in terms of its local singularities h=0.3 h=0.7 Ex: h(x0) = 0.3 implies f(x) is very rough around x0. h(x0) = 0.7 implies a “smoother” function around xo.

  10. Multifractal Formalism • Spectrum of singularities D(h) D(h) h • D(h) can be estimated from the statistical moments of the fluctuations. Legendre Transform

  11. Multifractal Spectra • Spectrum of scaling exponents t(q) and Spectrum of singlularities D(h) monofractal h multifractal h

  12. Slopes (q) (2) 1 2 4 5 6 7 q 3 -1 Multifractal Spectra Spectrum of scaling exponents Spectrum of singularities D(h) Df h hmax hmin

  13. The local singularity of f(x) at point x0 can be characterized by the behavior of the wavelet coefficients as they change with scale, provided that the order of the analyzing wavelet n > h(x0) Can obtain robust estimates of h(x0) using “maxima lines” only: Ta(x) i.e. WTMM It can be shown that Wavelet-based multifractal formalism(Muzy et al., 1993; Arneodo et al., 1995) CWT of f(x) :

  14. f(x) Structure Function Moments of |f(x+l) – f(x)| T[f](x,a) Partition Function Moments of |T[f](x,a)|  Partition Function Moments of |Ta(x)| (access to q < 0)  Cumulant analysis Moments of ln |Ta(x)| (direct access to statistics of singularities) WTMMTa(x)

  15. Two Examples • Landscape dissection • Planform topology of channelized and unchannelized paths (branching structure of river networks and hillslope drainage patterns) • Vertical structure of landform heterogeneity perpendicular to the river paths. • River bedform morphodynamics and sediment transport rates

  16. Width and Area Functions of a River Network # of channels intersected by a contour of equal flow length to the outlet # of pixels of equal flow length to the outlet Topology of river network Topology of the hillslope drainage paths and topology of river network

  17. Width and Area Functions of a River Network # of channels intersected by a contour of equal flow length to the outlet # of pixels of equal flow length to the outlet Topology of river network Topology of the hillslope drainage paths and topology of river network

  18. Area and Width Functions Walawe River, Sri Lanka (90x90m) A;2,000 km2

  19. Area and Width Functions Noyo River Basin, California, USA (10x10m) A;143 km2

  20. Walawe River Basin: deviation from simple scaling A(x) W(x) W(x) A(x) A rich multifractal structure is observed which is different for A(x) and W(x)

  21. Noyo River Basin: deviation from simple scaling A(x) W(x) A rich multifractal structure is observed which is different for A(x) and W(x)

  22. Noyo River Basin (10x10m; A;143 km2) c1 ; 0.77 c2 = 0.11 SR= 0.07 - 0.43 km • “Hillslope” path dominated • “smoother” overall than W(x) • Hillslope drainage dissection is s-s between scales 0.1 km – 0.5km • Statistics of the density of hillslope drainage paths strongly depend on scale c1 ; 0.46 c2 ;0.10 SR= 0.13 – 0.70 km • River network path dominated • “Rougher “overall” than A(x) • Channel network landscape dissection is s-s between scales 0.1 km to 0.7 km • Strong inntermittency (higher moments of pdf of channel drainage density has a strong dependence on scale) Pay attention not only to the average properties of landscape dissection but to higher moments

  23. Real River Networks Noyo River basin, CA (10x10m) A;1430 km2 Walawe River, Sri Lanka (90x90m) A;2,000 km2 South Fork Eel River, CA (1x1m & 10x10m) A;154 km2 C1=0.77 C2=0.11 C1=0.40 C2=0.05 C1=0.80 C2=0.05 A(x)

  24. Area Functions of Simulated Trees Peano Basin Not comparable to real networks Shreve’s random network model c1=0.5 c2=0 Stochastic S-S model with (a , b)=(1 , 2) c1=0.62 c2=0

  25. Conclusions on topology of drainage paths • Simulated river networks show different multifractal properties than real river networks. [s-s trees are monofractal with H = 0.5 – 0.65 while real networks are multifractal with H; 0.4 – 0.8]. • Differences between scaling properties of A(x) and W(x) depict differences in the branching topology of channelized vs. unchannelized drainage paths. • Deviation from monoscaling stresses the importance of the dependence on scale of higher order statistics of the branching structure.

  26. Implications for Network Hydrology? • Conjecture:Deviation from scale invariance in W(x), implies that the variability of the in-phase hillslope hydrographs entering the network depends on “scale” • ÞImplications for routing? scale-dependent convolution? geomorphologic dispersion? • Þ Implications for scaling of hydrographs?

  27. River Corridor Width Functions

  28. South Fork Eel River, CA Area = 351 km2

  29. River Corridor Width Function (D=5m)

  30. Questions • What is the statistical structure of RCW(x)? • Do physically distinct regimes exhibit statistically distinct signatures? • How can the statistical structure be used in modeling and prediction of hydrographs, sedimentographs and pollutographs across scales?

  31. SOUTH FORK EEL RIVER, CA: Hypsometric Profile

  32. River Corridor Width Function: South Fork Eel River 6 km 14 km 20 km 28 km 35 km 89 tributaries: (1 km2 – 150 km2)

  33. River Reach: 0-6 km

  34. SUMMARY OF RESULTS Right-Left asymmetry

  35. INTERPRETATION OF RESULTS More localized NL transport mechanism? More localized on L than R side? Smoother overall valleys? Presence of more terraces in R than L?

  36. Conclusions and Open Questions… • Hillslope “roughness” seems to carry the signature of valley forming processes; need to provide a complete hierarchical characterization. Do hillslope evolution models reproduce this structure? What is the effect on hillslope sediment variability of the higher order statistics of travel paths to streams?

  37. Pan-1 Pan-2 Pan-3 Pan-4 Pan-5 Experimental setup • Data Available: • Sediment accumulation series • Time series of bed elevation • Laser transects of bed elevation D50 = 11.3mm Discharge controlled here Channel Width = 2.75 m Channel Depth = 1.8 m D50=11.3 mm Diameter [mm] 100 1.0 • Discharge capacity: 8500 lps • Coarse sediment recirculation system located 55 m from upstream end.

  38. QUESTIONS • Do the statistics of sediment transport rates depend on “scale” (sampling interval or time interval of averaging) and how? • Does this statistical scale-dependence depend on flow rate, bed shear stress, and bedload size distribution (e.g., gravel vs. sand, etc.) • Do the statistics of sediment transport relate to the statistics of bedform morphodynamics and how? • What are the practical implications of all these?

  39. Sediment Transport Rates Nearest neighbor differences (S(t)) Accumulated series (Sc(t))

  40. VARIABILITY AT ALL SCALES Sc (t)= Accumulated sediment over an interval of 0 to t sec

  41. ANALYSIS METHODOLOGY: ADVANTAGES • Local analysis (as opposed to global, e.g., spectral analysis) • Can characterize the statistical structure of localized abrupt fluctuations over a range of scales • Wavelet-based multifractal formalism -- uses generalized fluctuations instead of standard differences (f(x) – f(x+dx)) • Can automatically remove non-stationarities in the signal both in terms of overall trends and in terms of low-frequency oscillations coming from dune or ripple effects • Can automatically remove noise in the signals and point to the minimum scale that can be safely interpreted • Can characterize effectively how pdfs change with scale with only one or two parameters

  42. SEDIMENT TRANSPORT RATES: Q = 5500 lps log2 Noise Variability levels off Scaling range C1=1.10 C2=0.10 15 min 1 min

  43. Q = 4300 lps log2 Noise C1=0.55 C2=0.15 Statistical Variability regime changes Scaling range 1 min 10 min

  44. Bed Elevation

  45. BED ELEVATION TEMPORAL SERIES: Q = 5500 lps Scaling range C1=0.70 C2=0.11 0.5 min 8 min

  46. BED ELEVATION TEMPORAL SERIES: Q = 4300 lps Scaling range C1=0.55 C2=0.05 12 min 1 min

  47. Inferences on Nonlinearity Basu and Foufoula-Georgiou, Detection of nonlinearity and chaoticity in time series using the transportation distance function, Phys. Letters A, 2002.

  48. Finite Size Lyapunov Exponent (FSLE) • FSLE is based on the idea of error growing time (Tr(d)), which is the time it takes • for a perturbation of initial size δ to grow by a factor r (equals to √2 in this work) • measure the typical rate of exponential divergence of nearby trajectory • δ(nr) size of the perturbation at the time nr at which this perturbation first exceeds (or becomes equal to) the size rδ • For an initial error δ and a given tolerance ∆ = rδ, the average predictability time Basu et al., Predictability of atmospherci boundary layer flows as a function of scale, Geophys. Res. Letters, 2002.

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